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%I #29 Apr 30 2018 11:53:21
%S 0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,
%T 0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,
%U 0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,0,1,0,0,0,0,1
%N The generalized Fibonacci word f^[5].
%C Shifted by 1, this is the binary sequence [(n+1)*alpha]-[n*alpha], n >= 1, where alpha = 1/(4+phi) has continued fraction [0,5,1,1,1,1,...]. Brown (1991, Theorem 3) shows that this is not fixed by any morphism 0 -> A, 1 -> B where A and B are finite binary strings. - _N. J. A. Sloane_, Sep 11 2016
%C In fact, none of the generalized Fibonacci words f^[i], and none of its shifts, are fixed by a morphism as soon as i>2. This follows from Allauzen's criterion for the f^[i]: they are Sturmian sequences with slope
%C alpha[i] = (i-phi)/(i^2-i-1) (see Ramirez et al., page 8),
%C so the algebraic conjugate of alpha[i] is (2i-1+sqrt(5))/(2i^2-2i-2) which lies in (0,1) for i>2. For the shifts of the f^[i] this follows from Yasutomi's work. - _Michel Dekking_, Apr 21 2018
%D S.-I. Yasutomi, On Sturmian sequences which are invariant under some substitutions, in Number theory and its applications (Kyoto, 1997), pp. 347-373, Kluwer Acad. Publ., Dordrecht, 1999.
%H W. W. Adams and J. L. Davison, <a href="http://www.jstor.org/stable/2041889">A remarkable class of continued fractions</a>, Proc. Amer. Math. Soc. 65 (1977), 194-198.
%H Cyril Allauzen, <a href="http://www.numdam.org/article/JTNB_1998__10_2_237_0.pdf">Une caractérisation simple des nombres de Sturm</a>, Journal de Théorie des Nombres de Bordeaux 10, no 2 (1998), 237-241.
%H P. G. Anderson, T. C. Brown, P. J.-S. Shiue, <a href="http://people.math.sfu.ca/~vjungic/tbrown/tom-28.pdf">A simple proof of a remarkable continued fraction identity"</a> Proc. Amer. Math. Soc. 123 (1995), 2005-2009.
%H T. C. Brown, <a href="http://dx.doi.org/10.4153/CMB-1991-006-4">A characterization of the quadratic irrationals</a>, Canad. Math. Bull, 1991, 34(1), 36-41.
%H José L. Ramírez, Gustavo N. Rubiano, and Rodrigo de Castro, <a href="http://arxiv.org/abs/1212.1368">A Generalization of the Fibonacci Word Fractal and the Fibonacci Snowflake</a>, arXiv preprint arXiv:1212.1368 [cs.DM], 2012.
%F Set S_0=0, S_1=00001; thereafter S_n = S_{n-1}S_{n-2}; sequence is S_{oo}.
%F From _Peter Bala_, Nov 19 2013: (Start)
%F a(n) = floor((n + 2)/(phi + 4)) - floor((n + 1)/(phi + 4)) where phi = 1/2*(1 + sqrt(5)) denotes the golden ratio.
%F If we read the present sequence as the digits of a decimal constant c = 0.00001 00000 10000 10000 01000 .... then we have the series representation c = sum {n >= 1} 1/10^floor(n*(phi + 4)). An alternative representation is c = 9*sum {n >= 1} floor(n/(phi + 4)) /10^n.
%F The constant 9*c has the simple continued fraction representation [0; 11111, 10, 10^5, 10^6, 10^11, ..., 10^A022095(n), ...] (see Adams and Davison).
%F Using this result we can find the alternating series representation c = 9*sum {n >= 1} (-1)^(n+1)*(1 + 10^A022095(3*n-1))/( (10^A022095(3*n-3) - 1)*(10^A022095(3*n) - 1) ).
%F The series converges very rapidly: for example, the first 10 terms of the series give a value for c accurate to more than 12 million decimal places. Cf. A005614, A221150 and A221151. (End)
%t fibi[n_, i_] := fibi[n, i] = Which[n == 0, {0}, n == 1, Append[Table[0, {j, 1, i - 1}], 1], True, Join[fibi[n - 1, i], fibi[n - 2, i]]];
%t fibonni[n_, i_] := fibonni[n, i] = Module[{fn, Fn}, For[fn = 0, True, fn++, Fn = fibi[fn, i]; If[ Length[ Fn] >= n + 1 && Length[Fn] > i + 3, Return[ Fn[[n + 1]]]]]];
%t a[n_] := fibonni[n, 5]; Table[a[n], {n, 0, 105}] (* _Jean-François Alcover_, Nov 21 2017, after _R. J. Mathar_ *)
%Y Cf. A003849, A005614, A221150, A221151. A022095, A230900.
%K nonn
%O 0
%A _N. J. A. Sloane_, Jan 03 2013
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